Séminaire Lotharingien de Combinatoire, 91B.45 (2024), 12 pp.
Sara C. Billey and Stark Ryan
Brewing Fubini-Bruhat Orders
Abstract.
The Bruhat order on permutations arises out of the study of
Schubert varieties in Grassmannians and flag varieties, which have
been important for over 100 years. The purpose of
this paper is to study variations on this theme related to
subvarieties of the spanning line configurations Xn,k as defined
by Pawlowski and Rhoades. These subvarieties are indexed by
Fubini words, or equivalently by ordered set partitions. Three
natural partial orders arise in this context; we refer to them as the
decaf, medium roast, and espresso orders. The decaf order is a
generalization of the weak order on permutations defined by covering
relations using simple transpositions. The medium roast order is a
generalization of the (strong) Bruhat order defined by the closure
relationship on the subvarieties. The espresso order is the
transitive closure of a relation based on intersecting subvarieties.
Many properties of Schubert varieties and Bruhat order extend to one
or more of the three Fubini-Bruhat orders. We examine some of the
many possibilities in this work.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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